For $\phi \in C^\infty_0(\mathbb{R})$, define the quantity $\langle u, \phi \rangle$ by:
$$ \langle u, \phi \rangle = \lim_{m \to \infty} \left[ \left(\sum_{k =1 }^m \phi\left(\frac{1}{k}\right) \right) - m\phi(0) - \phi'(0) \log m \right].$$
I would like to show that $u \in \mathcal{D}'(\mathbb{R})$. That is, I would like to show that $u$ defines a distribution on $\mathbb{R}$.
This is an example that comes from Friedlander and Joshi's Introduction to the Theory of Distributions. And I believe the original source of this example is Schwartz himself.
To prove the above fact, we will need to show that $\langle u , \phi \rangle$ is indeed finite. Then, we will have to show that $\langle u , \phi \rangle$ satisfies some type of semi-norm estimate. This means, for each compact $K \subset \mathbb{R}$, we need to find a $C > 0$ and an $N \in \mathbb{N}_0$ so that, for all $\phi \in C^\infty_0(\mathbb{R})$ with $ \text{supp } \phi \subseteq K$:
$$ |\langle u, \phi \rangle | \le C \sup \{|\partial ^i \phi (x)| : x \in K, 0 \le i \le N \}.$$
I am stuck on proving both finiteness and the seminorm estimate. To show finiteness, the only thought I've had so far is to the Mean Value Theorem. Then we get $\xi_k \in \left[0, \frac{1}{k}\right]$ so that
$$\phi\left(\frac{1}{k}\right) - \phi(0) = k \phi'(\xi_k).$$
This allows us to rewrite $\langle u , \phi \rangle$ as
$$\langle u, \phi \rangle = \lim_{m \to \infty} \left[ \left(\sum_{k =1 }^m k\phi'(\xi_k) \right) -\phi'(0) \log m \right].$$
But I am not sure if this achieves anything. Hints or solutions are greatly appreciated
The hint would be to write $$\phi(1/k) = \phi(0)+\frac{\phi'(0)}{k}+\frac{\phi^{(2)}(\xi_k)}{2k^2},\quad \xi_k\in(0,1/k).$$
After some manipulations you will obtain that $u$ is indeed a distribution of at most second order.